Existence of constant scalar curvature Kaehler cone metrics, properness and geodesic stability

We show that the existence of constant scalar curvature K\"ahler (cscK) metrics with cone singularities is equivalent to the properness of log $K$-energy. We also prove their equivalence to the geodesic stability. They are extensions of the solution of the properness conjecture and Donaldson's geodesic stability conjecture of the cscK problem for smooth K\"ahler metrics by Chen-Cheng to the setting of cscK cone metrics. One applications of our main results is that we introduce and construct singular cscK metrics with possible degeneration in big cohomology class. As another application, we also prove both openness and approximation property for the path of cscK cone metrics, which are paralleling to Donaldson's continuity method through K\"ahler-Einstein cone metrics in the resolution of Yau-Tian-Donaldson conjecture for Fano K\"ahler-Einstein metrics.